Self-similarity study based on the particle sizes of coal-series diatomite

Coal-series diatomite (CSD) is widely distributed in China and has poor functional and structural properties and exhibits limited utilization of high value-added materials, resulting in a serious waste of resources and tremendous pressure on the environment. Moreover, due to differences in the mineralogical characteristics of CSD, different particle size scales (PSSs) have different functional structures and exhibit different self-similarities. In this study, we took CSD as the research object and PSS as the entry point and carried out a self-similarity study based on gas adsorption and an image processing method to illustrate the microstructures and self-similarities of different PSSs. The results showed that the pore structure of the CSD was dominated by mesopores and macropores and basically lacked micropores. The fractal dimensions were calculated with the Frenkel-Haisey-Hill (FHH) model and Menger model, and the DF1 values for − 0.025 mm and − 2 mm were 2.51 and 2.48, respectively, and the DM1 values were 3.75 and 3.79, respectively, indicating that the mesopore structure of the fine PSS was complex, whereas macropores were present in the coarse PSS. MATLAB was programmed to obtain grayscale thresholds, binarized images, grayscale histograms, three-dimensional (3D) reconstruction images and box dimensions, which enabled us to observe the microstructures and self-similarities of the CSD. Self-similarity studies based on particle sizes are very important for functional application of CSD.Please note that article title mismatch between MS and JS we have followed MS, kindly check and cofirm.Yes, I have checked and confirmed.Kindly check and confirm corresponding author mail id are correctly identified.Yes, I have checked and confirmed.


Preparation of samples with different PSSs
The CSD was taken from the Yong'an mining area in Heilongjiang Province, the particle sizes were less than 2 mm, and the diatom particles were incomplete.The chemical composition is shown in Table 1, and the main component was SiO 2 (59.22%), which also contained impurities such as Al 2 O 3 , Fe 2 O 3 , K 2 O, Na 2 O, CaO, MgO, etc.Meanwhile, the loss on ignition (LOI) was 10.50%.The mineral composition is shown in Fig. 1, in which the main components were opal and quartz, followed by feldspar, mica, montmorillonite, haematite, magnetite and other minerals, and the content of opal was approximately 12.39%, which accounted for approximately one-eighth of the total mineral content.

Characterization methodology
Based on previous qualitative analyses of CSD, the chemical compositions of the samples were quantitatively analysed with volumetric and gravimetric methods 37 ; the mineral compositions and crystal structures of the CSDs were analysed with a D8 ADVANCE X-ray diffractometer (XRD, Bruker, Germany) 38 ; and the SSAs and pore structures of CSDs with different PSSs were measured with a 3H-2000PMV autospecific surface area analyser (Beishide Instrument S&T.(Beijing) Co., Ltd., China) 39 .First, the total pore volumes (V T ) of the CSD samples were calculated with the BET model based on low-temperature N 2 isothermal adsorption and desorption data.Second, the average pore diameters (D a ) were calculated based on the V T and SSA.Finally, the distributions of the SSAs and pore diameters for CSDs with different PSSs were analysed with the BJH model 40,41 .
Based on the data from the desorption branches of the N 2 isothermal adsorption-desorption curves, the FHH model [42][43][44] suggested that the gas adsorbed on the fractal surface can be represented as Eq.(1).
where p is the equilibrium pressure, V is the volume of adsorbed N 2 at the equilibrium pressure, V 0 is the volume of adsorbed N 2 in the monolayer, p 0 is the saturated N 2 pressure, lnV is linearly fitted to ln[ln(p 0 /p)], and A is a constant related to the fractal dimension (D') and the adsorption mechanism.When the adsorption mechanism is capillary condensation, D' can be expressed as in Eq. (2).
The pore structure of the CSD was analysed with the Menger model 45 , and Eq. ( 3) was obtained based on the pore volume V p (r) = R 3 -V(r).
The fractal dimension of a self-similarity pore structure was translated into a measure of the N 2 pressure and the pore volume, which were derived directly from the slope of a double logarithmic plot of dVp/dp vs. p 31 .
A Sigma 300 scanning electron microscope (SEM, Zeiss, Germany) was used to measure the pore structures and analyse the self-similarity fractal characteristics of the CSDs with different PSSs 32 .An image that had been stored within the computer, which was composed of pixel points of size k*, was considered.Since the SEM image contained parametric information that could be mistaken for particles and pores during image processing, thus affecting the accuracy of the image, it was necessary to crop the image, subtract the boundaries and leave an accurate image that contained only the CSD image without text, and the image pixels for this experiment were 2837 × 2121.The image was binarized so that each pixel point was either black or white.For this study, the image in jpg format was converted to a grayscale image with the rgb2gray function in MATLAB software.It was first binarized with the optimal segmentation thresholding method to obtain a binarized image with background pixels as particles (white) and target pixels as pores (black).The optimal threshold segmentation method 46,47 originated from fractal theory in image processing, and the basic principle was to determine a grayscale threshold.Then, the relationship between the size of the pixel grayscale value and the threshold value in the image was determined for image segmentation.Using this method, the boundary contour of the substance was clearly defined, and the image was converted into black and white so that the information of the image was stored in a matrix, with each point representing a pixel of the image.
In binarization of the images, some of the particles were mistaken for apertures when the threshold value was too large; thus, the optimal threshold value was determined to ensure that most of the particles were defined in terms of their contours.The following binarization algorithm with an optimal segmentation threshold was used: www.nature.com/scientificreports/ (1) The minimum and maximum grayscale values f min and f max in the image were found and denoted the initial value of the threshold, as in Eq. ( 5).
(5) (2) The digital image was divided into foreground and background parts with the threshold value derived in the previous step, and the average grayscale values f A and f B of the two parts were found with Eqs. ( 6) and (7).
where f(i,j) is the grayscale value of the (i,j) point corresponding to the planar image and w(i,j) is the corresponding weight coefficient, which is generally taken as w(i,j)=1.
The SEM image was first converted into a grayscale image, in which the darker grayscale region was the pores, and the grayscale value was analysed.Then, the threshold l was obtained with the above algorithm and Eq. ( 8) to extract the microstructural information in the SEM image.
where l is the threshold value obtained with the optimal segmentation thresholding algorithm.A program in MATLAB was used to convert the above binarized image into a data file, in which the number of rows and columns corresponded to the binarized image, and each frame of data was assigned a value of 1 or 0 depending on the colour of the pixel point to which it corresponded (Fig. 6); 1 indicated that the corresponding pixel point in the binarized image was white and 0 indicated that it was black.
Self-similarity fractal features are objective characteristics of many natural things and phenomena, and physical fractals that exist in nature tend to show scale and randomness; i.e., they only show fractal features statistically at a particular scale.Characterisation of the fractal for a research object depends on the characteristics of the object as well as the purpose of the research, and different descriptive methods can be used to compute the fractal dimensions, such as the Hausdorff dimension 48 and similarity dimension 49 .However, for many fractals, both of these dimensions are difficult to compute; in practice, the box dimension 32,50 is generally used, and one of its equivalent formulas is shown in Eq. (9).
where C is a constant; k is the size of the box; and N k indicates the number of boxes of size k needed to cover all foreground images.In the calculation, the data file is divided into blocks, each of which is a square with side lengths containing pixels of number n.The number of blocks containing 0 is denoted as N kn .If the size of a pixel point is k*, the length of a block with n rows and columns is k n = nk*.The corresponding number of boxes can be obtained by using k n as the edge length for block division.
In 3D reconstructions of SEM images with MATLAB software [51][52][53] , the first step was to collect and process the 3D image data, and then, according to the imaging principle, calculate the action of all pixels on the light to obtain the two-dimensional (2D) projection image and construct the 3D body reconstruction fragments.The 3D display effect was set up to complete the 3D reconstruction of SEM images.The body projection method was mostly used to calculate the projection of an object on a plane 54,55 .By projecting points on the surface of the 3D object, i.e., the 3D data points after preprocessing, along the parallel lines to the 2D plane, and by transforming the angle between the parallel lines and the 3D object so that all of the 3D data points were projected on the 2D surface, the projection principle is shown in Fig. 13b.The conversion from a 2D to 3D display was realized by continuously projecting the map.The operation was simplified by the MATLAB software, and the projection ( 6)

Results and discussion
Pore structure analyses Fig. 3 shows the results of N 2 isothermal adsorption and detachment of the CSD after drying, mixing, condensation, and crushing to particle sizes of less than 2 mm.As shown in Fig. 3a-f, the CSD had different colours, such as black-grey, dark-grey, and white-grey, which was attributed to nonuniformity of the water contents of the samples.As shown in Fig. 3g, the N 2 isothermal adsorption-desorption isotherms of the CSDs exhibited type IV isothermal curves, and there were obvious hysteresis loops in the relative pressure (p/p 0 ) range 0.6-0.9,indicating the presence of a typical mesoporous structure in the CSD 56 .The isothermal adsorption desorption curves did not show high plateaus at high relative pressures (p/p 0 > 0.9), indicating the presence of macropores in the CSD.
Little N 2 was adsorbed at low relative pressures (p/p 0 < 0.1), indicating that the microporous structure in the CSD was not developed.The geometric nonuniformity of the pore structures determined that a single pore diameter did not accurately characterize the real pore distribution, and a reasonable model was needed to analyse the pore diameter distribution.Fig. 3h,i shows the pore diameter distribution curves and the SSA vs. pore diameter distribution curves of CSDs analysed by the BJH model.As shown in Fig. 3h, there was an obvious peak for dV/ dD in the range of pore diameters from 0 nm to 10 nm, indicating that the pore distribution in this range was more concentrated.As shown in Fig. 3i, the dS/dD plot of the CSD had an obvious peak in the pore diameter range 2 nm to 5 nm, indicating that the pore distribution in this range was also more concentrated.Moreover, www.nature.com/scientificreports/when the pore diameter was larger than 10 nm, a part of the total SSA distribution curve for the CSD still existed in the pore diameter range, but the proportion of SSA was smaller than that of the SSA distribution for the pore diameters in the range 2-10 nm, which suggested that the SSA of CSD was mainly determined by the mesopores with smaller diameters.As shown in Fig. 3j, the SSA curve showed a good fit with a correlation coefficient (R 2 ) of 0.9999, a slope (a) of 0.0838, an intercept (b) of 0.0012, and a BET constant (c) of 74.1 in the range of relative pressures 0.04 < p/p 0 < 0.32.Using Eq. ( 10), the SSA was calculated as 51.2128 m 2 •g −1 .
where p is the pressure after adsorption equilibrium; p 0 is the saturated vapour pressure of the adsorbent at the adsorption temperature; V m is the saturated adsorption capacity for a single layer of the adsorbed material; 4.35 is the area occupied by a single-molecule layer of 1 mL N 2 under standard conditions; and V is the amount of adsorbate on the adsorbent at the adsorption equilibrium pressure.Fig. 4a, b shows the N 2 isothermal adsorption and desorption data for CSDs with different PPSs.The isothermal adsorption and desorption curves, pore diameters and SSAs of the CSDs at 9 PPS reflected similar changes in the pore structures.As the particle sizes of the CSDs decreased, the corresponding hysteresis loop expanded, the pore diameters decreased, and the SSA increased.The isothermal adsorption and desorption curves of CSDs with different PPSs were indicated type IV isothermal curves, and were hysteresis loops within the range of relative pressures p/p 0 = 0.6-0.9, which indicated that there was a typical mesoporous structure in the CSD.An obvious hysteresis loop was observed in the isothermal adsorption desorption curve of the − 0.025 mm sample, indicating that the mesopores for the CSD with a fine PPS were obviously more abundant, which was attributed to a reduction in gangue minerals, which weakened the agglomeration of CSD particles and resulted in a richer pore structure.The isothermal adsorption desorption curves did not show high plateaus at high relative pressures (p/p 0 > 0.9), suggesting that CSD contained macropores.Little N 2 was adsorbed at low relative pressures (p/p 0 < 0.1), indicating that the microporous structure of the CSD was not developed.Moreover, N 2 adsorption at relative pressures p/p 0 = 0.98 increased from 55.8441 cm 3 •g −1 for − 2+1 mm to 75.3813 cm 3 •g −1 for − 0.025 mm, and N 2 adsorption for − 0.025 mm was significantly greater than that for the other PPSs, showing an overall increase with decreasing particle sizes.The dV/dD and dS/dD for the CSDs with different PPSs had obvious peaks in the range of pore diameters from 0 nm to 10 nm, indicating that the distribution of pores was more concentrated in this range.Moreover, when the pore diameter was larger than 10 nm, a part of the total SSA distribution curve for the CSD was still present in the range of pore diameters at different PPSs, but the proportion of the SSA accounted for was smaller than that for the SSA distribution for pore diameters pf 2 nm to 10 nm, indicating that the SSAs of the CSDs with various PPSs were mainly determined by the mesopores with smaller diameters.In addition, the R 2 values of the SSA fitted curves at different PPSs in the range of relative pressures 0.04 < p/p 0 < 0.32 were greater than 0.9 (Fig. 4b), indicating that the fits were better, the SSA increased from 26.8294 m 2 •g −1 for − 2+1 mm to 47.0258 m 2 •g −1 for − 0.025 mm, and the SSA of − 0.025 mm was significantly larger than those of the other PPSs, showing an overall increase with decreasing particle size.
Table 2 and Fig. 5 show the main pore structure parameters of CSDs analysed by the gas adsorption method (N 2 ) at different PPSs, in which N 2 adsorption was positively correlated with the SSA, and the SSA decreased from 47.0258 m 2 •g −1 at − 0.0258 mm to 4.6050 m 2 •g −1 at − 0.5+0.25 mm.The hysteresis loops of the isothermal adsorption and desorption isotherms for CSDs with different PPSs in the range of − 0.5 mm generally showed gradual decreases with increasing PSS, indicating that the V mes of the CSD basically decreased after grading (Fig. 4a, b).Moreover, the V T values of the CSDs with different PSSs in the range of − 0.5 mm decreased with increasing PSS, and the V T reached a maximum value of 0.1407 cm 3 •g −1 at − 0.025 mm, which was related to the embedded particle sizes of the gangue minerals in the CSDs with different PPSs; as the PSS increased, the pore diameter increased, which increased the pore volume.The t-plot method was used to analyse the microporous structures of CSDs with different PSSs, and the microporous volumes (V mic ) were close to 0 cm 3 •g −1 , indicating that there were basically no micropores in the CSD.Combined with the N 2 isothermal adsorption desorption isotherms and pore diameter distribution curves, the pore channels of the CSD were mainly multilevel pore structures composed of mesopores and macropores, and basically, there were no micropores.
The formation of the CSD pore structure was attributed to the 3D constitutive units comprising SiO 4 tetrahedra, which exhibited different pore diameters, pore volumes, and SSAs due to plugging and encapsulation of the companion minerals.Fig. 6 shows the 3D spatial surface maps of the pore diameters -pore volumes -SSAs for CSDs with different PPSs, which showed that there was a constraint relationship between the pore diameters, pore volumes, and SSAs, and smaller pore diameters and larger pore volumes generated larger SSAs.

Fractal characterization with the gas adsorption method
There were two distinct linear regions of the isothermal adsorption-desorption isotherms for CSDs with different PPSs, which were located in the relative pressure ranges p/p 0 = 0-0.35and 0.35-1.0(Fig. 4).With the gas adsorption method, the N 2 isothermal desorption data for CSDs were different PPSs were fractally fitted with the FHH model, and the FHH fits are shown in Fig. 7.The linear fitting equations, fractal dimensions (D F1 and D F2 ), and fitting correlation coefficients (R 2 ) are given in Table 3 for the pore channels in CSDs with pore diameters less than 50 nm and different PPSs.The results showed that the CSD had good fits and different fractal characteristics in both regions, with R 2 values greater than 0.93.D F1 and D F2 reflect the inhomogeneity of the surface roughnesses and mesopore volumes of the CSD at different PPSs, respectively, and quantitatively characterize the nonhomogeneity and complexity of the pore structure.The fractal dimension influences the interaction of porous mineral surfaces with other substances during physicochemical processes such as adsorption, adhesion and surface diffusion.Table 3 shows the D F1 and D F2 values calculated from the fitting results in Fig. 7, with D F1 (10)   The number of mesopores with small pore diameters in the fine PSS was greater, the distribution of the mesopore apertures was narrower, the number of mesopores with large pore diameters in the coarse PSS was greater, and the distribution of mesopore apertures was wider.The increases in D F1 with decreasing particle sizes indicated that the surface roughness of the CSD increased with decreasing particle sizes, which was consistent with the change rule for SSAs with different PSSs in Table 2 and Fig. 5.In addition, the D F2 also increased with decreasing particle sizes, and the D F2 reached the lowest value of 2.6799 at − 0.025 mm.Moreover, Fig. 7 shows the results of Menger model fitting for the fractal features of the macroporous structure of CSDs at different PSSs.Table 3 shows the linear fitting equations, fractal dimensions (D M1 and D M1 ) and R 2 values for the pore channels of CSDs with pore diameters larger than 50 nm.As shown in Table 3, the D M  2. Pore structure parameters of CSDs with different PSSs after low-temperature N 2 adsorption.SSA is the specific surface area, S mic is the specific surface area of the micropores, V T is the total pore volume of N 2 adsorption, V mic is the micropore volume, V mes is the mesopore volume, D a is the average pore diameter, D mes is the average mesopore diameter, and D mmes is the most probable mesopore diameter.

Fractal characterization of the image processing method
Fig. 8 shows SEM images of the − 2 mm and − 0.025 mm samples.As shown in Fig. 8a, the mineral composition in the CSD was complex, with a high content of gangue minerals, diatomite surface covering and pore channel plugging with clay minerals, and poor pore structure characteristics.As shown in Fig. 8b, the diatoms in the fine PSS of the CSDs had complete morphologies, more 3D pore channels, and better pore structure characteristics.In comparison, the diatoms are mainly enriched in the fine PSSs of the CSDs and showed better pore structure characteristics.SEM image analyses and self-similarity fractal characterizations of the CSDs with different PSSs were carried out via image processing (Fig. 9).As shown in Fig. 10, in the SEM images of the CSDs with different PSSs, the selected images had more obvious fractal features and thus were more representative.The black areas in the image were the pore channels, and the white areas were the particles and skeletons.As seen in the RGB images, the CSD was composed of rounded diatom particles, which were structured and connected together by the outer rings.Due to this structural feature, there were many pore channels of different diameters between and within the diatom particles.As the particle sizes of the CSD decreased, the diatom surface was covered, and pore channels were blocked by fewer gangue minerals, exposing more pore channels.At − 0.025 mm, the macropores between the diatom particles gradually disappeared, and the internal mesopores were exposed.The grayscale thresholds of the CSDs with different PSSs were obtained from algorithms (6-7) as 106 (− 2+1 mm), 113 (− 1+0.5 mm), 101 (− 0.5+0.25 mm), 119 (− 0.25+0.15mm), 151 (− 0.15+0.075mm), 137 (− 0.075+0.045mm), 100 (− 0.045+0.038mm), 91 (− 0.038+0.025mm), and 144 (− 0.025 mm).The grayscale images processed with the Eq. ( 8) algorithm are shown in binary images, with the background pixels as particles (white) and the target pixels as pores (black).
According to Eq. ( 9), a plot of lg(1/k) vs.As shown in Fig. 11, the SEM images of the CSDs with different PSSs had better fractal characteristics after binarization, and the correlation between the experimental data points and the fitted straight line was high.Thus, the microstructural changes of the CSDs with different PSSs were described by the fractal box dimension.Combined with the SEM images in Fig. 10, the changes in fractal features were divided into two stages: (1) from the perspective of D1 to D4, the variations in the box dimensions of the coarse PSS were not significant and relatively large due to the clogging and encapsulation of the mesopores with gangue minerals and the presence of structures such as cracks, concavity and convexity, and steps; (2) from the perspective of D5-D9, the box dimensions of the fine PSS increased with decreasing particle sizes, which was consistent with N 2 isothermal adsorption-desorption, FHH modelling, and SEM imaging.However, the overall box dimensions of D5-D9 were smaller than those of D1-D5.This is because the mineralogical characteristics of the CSD had a significant impact on the box dimension, while the effects of particle gaps and gangue mineral particles on the box dimension were relatively small for fine PSSs.Therefore, the fractal box dimension characterized the changes in the self-similar pore structure and the changes in the pore distribution after the disappearance of structural properties for CSDs at different PSSs discussed in this study.
To visualize the self-similarity 3D pore structures of the CSD, 3D SEM images of the different PSSs were reconstructed.(1) Based on the SEM grayscale image, the grayscale histogram was created by the imhist function in MATLAB software (Fig. 12).The horizontal axis of the histogram represents the grayscale value, the vertical axis represents the pixel, and each vertical line represents the number of pixels contained under the grayscale value, i.e., the frequency of each grayscale value appearing in the whole SEM image of the CSD.The grayscale values ranged from 0 to 255, and the blacker the image was, the smaller the grayscale value, and the brighter the image was, the larger the grayscale value; i.e., the pixels with grayscale values of 0 were black and represented apertures in the SEM image, the pixels with grayscale values of 255 were white and represented particles, and the other pixels were between black and white.(2) The preprocessed image was added to the search path of the MATLAB software, and then the image information was extracted by the imread function.The pixels of the CSD image used in this experiment were 2837 × 2121, the pixels of the image were used as the length and width, the grayscale value was taken as the height to obtain the 3D dataset, and the data were stored in a CAT file for subsequent use.(3) The acquired 3D data were projected on the apparent plane with the body projection method, i.e., the acquired 3D body data of the SEM image were projected onto the plane that was observed visually    www.nature.com/scientificreports/according to the projection method, and then the projected equivalent surface was drawn with the isosurface function.(4) The colour, shadow and display effect of the image were set to make the 3D model more intuitive, and comprehensive 3D reconstructions of the CSDs were achieved at different PSSs.The final 3D reconstruction is shown in Fig. 13.The reconstructed 3D image provided a good 3D display, the undulations of the self-similarity structures of the CSDs were clearly observed, and the sizes, shapes and distributions of the particles and pores were all consistent with the actual pore structure characteristics of CSDs at different PSSs.

Conclusions
CSD comprises coal-series waste through open piling and backfilling of mining areas, which causes a serious waste of resources and great pressure on the environment.In this study, we took CSD as the research object and PSS as the entry point and conducted a self-similarity study based on an analytical gas adsorption method, an image segmentation method, an optimal threshold method and a 3D reconstruction method to illustrate the self-similarity 3D pore structures of CSDs at different PSSs.
The main chemical component of the CSD was SiO 2 (59.22%), and the LOI was 10.50%; the main mineral component was opal, with a content of approximately 12.39%, which accounted for approximately one-eighth of the total mineral content.Opal was the main constituent mineral of the diatom skeleton, with a chemical composition of SiO 2 •nH 2 O, and formation of the pore structure (diatom skeleton) was attributed to the 3D constituent unit built by SiO 4 tetrahedra.The self-similarity pore structure was poorly characterized due to the low SiO 2 content and the complexity of the associated minerals in the CSD.
The pore structures of the CSDs with different PSSs were dominated by mesopores and macropores, with basically no micropores.The SSA and V mes values were positively correlated with the amount of N 2 adsorbed and negatively correlated with the PSS in general.The fractal dimensions calculated with the FHH model were 2.51/2.68 for − 0.025 mm and 2.48/2.64 for − 2 mm, suggesting that the mesopore structure of fine PSS was complex, whereas macropores were predominantly present in the coarse PSS.This was attributed to clogging and encapsulation of gangue minerals as well as the presence of structures such as cracks, concave and convex cracks, and steps that gave the CSD different self-similarity pore structural characteristics at different PSSs.
The grayscale thresholds, binarized images, grayscale histograms, 3D reconstruction images, and a series of box dimensions calculated from the SEM images of CSD at different PSSs in the image processing method enabled intuitive indications of the microstructure of the CSDs as well as self-similarity features.Meanwhile, the reconstructed 3D image provided a good 3D display, the undulations of the self-similarity functional structures of the CSDs were clearly observed, and the sizes, shapes, and distributions of the particles and pore channels matched the pore structures of the actual CSDs.Self-similarity studies based on particle sizes are important for the functional application of CSDs.

Figure 3 .
Figure 3.Samples of raw soil (a, b), samples after drying (c, d), samples after mixing and shrinking (e), samples after crushing and shrinking to − 2 mm particle sizes (f), isothermal adsorption desorption curve (g), pore diameter distribution curves of the BJH Model (h), SSA distribution curves of the BJH Model (i), and SSA fitting curve of the BET Model (j).
lgN k was subject to least squares fitting in the double logarithmic coordinate plane, and the slope of the resulting straight line was the number of box dimensions.The binary images were imported into the written MATLAB program, and the fitting results are shown in Fig. 11.The N k values of the CSDs with different PSSs are shown in Table4.

Figure 9 .
Figure 9. SEM image processing and analyses (grayscale histogram, binary processing, 3D reconstruction and fractal dimension) were performed with MATLAB.

Table 1 .
Contents (wt.%) of the major chemical and mineral components (by pattern fitting and Rietveld refinement) of the CSD.

Table 3 .
Fractal dimension (D F = 3 + A, D M = 4 + A) of CSDs with different PSSs derived with the FHH model and Menger model.

Table 4 .
N k for different PSSs of the CSDs.